Approximate Stream Reasoning with Incomplete State Information - - PowerPoint PPT Presentation

Approximate Stream Reasoning with Incomplete State Information

Fourth Stream Reasoning Workshop, Link¨

• ping, Sweden

Daniel de Leng

Artificial Intelligence and Integrated Computer Systems Department of Computer and Information Science Link¨

• ping University, Sweden

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Metric Temporal Logic Progression-based Runtime Verification

Introduction

1 Introduction 2 Stream Reasoning with Incomplete Information 3 Progression Graph-Based Progression 4 Summary

Daniel de Leng Link¨

• ping University

2/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Metric Temporal Logic Progression-based Runtime Verification

Introduction

Consider runtime verification for checking whether an agent is behaving in a safe manner. Example (Safety) “A robot in an unsafe state should return to a safe state within 10 seconds” Motivation: Incomplete information!

Daniel de Leng Link¨

• ping University

3/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Metric Temporal Logic Progression-based Runtime Verification

Metric Temporal Logic

We use Metric Temporal Logic (MTL) as a language for describing temporal rules that must hold. Definition (MTL syntax) The syntax for MTL is as follows for atomic propositions p ∈ Prop, temporal interval I ⊆ (0, ∞), and well-formed formulas (wffs) φ and ψ: p | ¬φ | φ ∨ ψ | φ UI ψ where I and ♦I are syntactic sugar for ‘always’ and ‘eventually’.

Daniel de Leng Link¨

• ping University

4/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Metric Temporal Logic Progression-based Runtime Verification

Progression-based Runtime Verification

Progression is an incremental syntactic rewriting procedure introduced by Bacchus and Kabanza (1996, 1998): MTL Formula + Complete State + Delay ⇒ MTL Formula φ0 = (¬p → ♦[0,10]p), s = {¬p} , ∆ = 2 φ1 = ♦[0,8]p ∧ (¬p → ♦[0,10]p)

Daniel de Leng Link¨

• ping University

5/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Incomplete States and Streams Progression Graphs

Stream Reasoning with Incomplete Information

Problem: How to perform progression with incomplete states? General idea: Apply model counting

Daniel de Leng Link¨

• ping University

6/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Incomplete States and Streams Progression Graphs

Incomplete States and Streams

Important assumptions: We keep a constant delay value (∆) and omit it from here on; An incomplete state s is a disjunctive set of complete states; A (piecewise) incomplete stream ρ is a sequence of incomplete states; We assume we have a probabilistic model of a stream denoted by a state universe Sn for every time-point n.

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• ping University

7/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Incomplete States and Streams Progression Graphs

Progression Graphs

A progression graph encodes formulas and their progressions into a graph G(χ) = (χ, V , E) such that vertices represent formulas; χ ∈ V represents the graph source formula; and labelled edges (φ, ψ, s) ∈ E iff PROGRESS(φ, s) = ψ.

⊤ ⊥ p {p} ∅ ◇[0,1] p {p} ∅ ◇[0,2] p {p} ∅ ◇[0,3] p {p} ∅ ◇[0,4] p {p} ∅ ◇[0,5] p {p} ∅

Daniel de Leng Link¨

• ping University

8/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Incomplete States and Streams Progression Graphs

Progression Graphs

Progression graphs Gn(χ) = (χ, V , E, mn) carry probability mass: m0(χ) = 1.0 (Initialization) mn(v) =

• (v′,v,s)∈E
• mn−1(v′) Pr[Sn = s |

sn]

• Theorem (Soundness)

Given a progression graph Gn(χ) and a stream ρ: lim

n→∞ mn(⊤) = Pr [

ρ, t0 | = χ] , lim

n→∞ mn(⊥) = Pr [

ρ, t0 | = χ] .

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• ping University

9/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression

Progression Graph-Based Progression

Example (Ship Stabilisation) Suppose we have an autonomous ship with a landing deck. The ship attempts to stabilise itself according to the rule: (¬p → (♦[0,5][0,3]p)) “Whenever the ship is unstable (¬p), the ship will be stable (p) for a consecutive period of 3 minutes, within 5 minutes from having become unstable.“

Daniel de Leng Link¨

• ping University

10/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression Daniel de Leng Link¨

• ping University

11/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression Daniel de Leng Link¨

• ping University

11/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression Daniel de Leng Link¨

• ping University

11/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression Daniel de Leng Link¨

• ping University

11/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression

Incomplete Information

Example (Ship Stabilisation (Cont’d)) Suppose we are no longer able to measure unambiguously whether the ship is stable. Continue progression, and assume 90% stable, 10% unstable.

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• ping University

12/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression Daniel de Leng Link¨

• ping University

13/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression Daniel de Leng Link¨

• ping University

13/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression Daniel de Leng Link¨

• ping University

13/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression Daniel de Leng Link¨

• ping University

13/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression Daniel de Leng Link¨

• ping University

13/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression Daniel de Leng Link¨

• ping University

13/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression Daniel de Leng Link¨

• ping University

13/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression Daniel de Leng Link¨

• ping University

13/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression

Example: Ship Stabilisation

Example (Ship Stabilisation (Cont’d)) After 10 minutes, despite incomplete sensor readings, we know: Pr[ ρ, t0 | = (¬p → (♦[0,5][0,3]p))] ≥ 0.212680, right now based on m10(⊥), regardless of future readings.

Daniel de Leng Link¨

• ping University

14/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression

Approximate Progression

Approximate progression allows us to trade precision for speed and vice-versa:

1 Institute a MAX AGE for formulas; 2 Limit the size of the graph by setting a MAX NODES bound.

We may drop nodes with probability mass, thereby leaking some probability mass over time.

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• ping University

15/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression

Methods to reduce the graph size: MAX AGE and MAX NODES.

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16/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression

Performance penalty: MAX AGE = 3

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17/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary Complete Information Incomplete Information Approximate Progression

Precision penalty: MAX NODES = 5

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• ping University

18/19

Introduction Stream Reasoning with Incomplete Information Progression Graph-Based Progression Summary

Summary

Summary:

1 Classical progression assumes complete states; 2 We extended progression to handle incomplete states; 3 Progression graphs allow us to implicitly keep track of traces; 4 Approximation offers a trade-off between precision and speed.

Many interesting extensions possible!

Daniel de Leng Link¨

• ping University

19/19